In linear elastic behaviour unloading:

Follows the same straight line back to origin

Follows a curve with higher stiffness

In inelastic behaviour, unloading follows:

A line parallel to the initial elastic slope, leaving permanent deformation

A line steeper than the initial slope that returns to origin

A non-linear elastic structure (e.g. rubber band):

Returns along the same curve to the origin with no permanent deformation

Linear/non-linear versus elastic/inelastic differ in that:

Linear/non-linear describes curve shape; elastic/inelastic describes recovery on unloading

2071 Magh · 4 marks

Elastic, inelastic, linear and non-linear behaviour

Short answer

Linear elastic: straight force-displacement line through origin, unloading returns on same path, no permanent deformation. Non-linear elastic: curved path but unloading returns to origin. Inelastic: unloading parallel to initial slope leaves residual deformation, e.g. steel after yield. Non-linear: stiffness or slope changes with load, so superposition is not valid.

Explain with suitable force-displacement diagrams the elastic, inelastic, linear and non-linear behaviour of structure.

The behaviour of a structure is described by its force-displacement (load-deflection) curve during loading and unloading.

 F |      /             F |         __--
   |     /                |      _-'
   |    /  loading =      |    /   loading =
   |   /   unloading      |   /    unloading
   |  /                   |  /     (same curve)
   | /                    | /
   |/                     |/
   +----------- D         +------------ D
  (a) Linear elastic     (b) Non-linear elastic

 F |      _____         F |         __--
   |     /    /           |      _-'
   |    /    / unloading  |    /    stiffness
   |   /    /  parallel   |   /     (slope) changes
   |  /    /   to initial |  /      with load
   | /    /    slope      | /
   |/    /                |/
   +----+------ D         +------------ D
     residual
  (c) Inelastic          (d) Non-linear

1. Linear elastic: The load-deflection curve is a straight line through the origin. On unloading, the structure returns to its original shape along the same line. Example: a steel beam under working loads.

2. Non-linear elastic: The curve is not straight, but unloading follows the same curve back to the origin, so there is no permanent deformation. Example: a rubber band, or a cable whose sag changes under load.

3. Inelastic: Beyond the elastic limit, the loading and unloading paths differ. Unloading follows a line parallel to the initial elastic slope, leaving a residual (permanent) deformation. Example: steel stressed beyond yield.

4. Non-linear: The stiffness (slope of the curve) changes with load. This arises from material non-linearity (yielding, cracking) or geometric non-linearity (large displacements). The principle of superposition is not valid.

Linear and non-linear describe the shape of the curve; elastic and inelastic describe whether the deformation is recovered on unloading.

✓ Answer checked by Claude Opus 5.5 (high effort)

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